Relativity - The Special and General Theory Part 7

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab. When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping. We next lay four of these little rods on the marble slab so that they const.i.tute a quadrilateral figure (a square), the diagonals of which are equally long. To ensure the equality of the diagonals, we make use of a little testing-rod. To this square we add similar ones, each of which has one rod in common with the first. We proceed in like manner with each of these squares until finally the whole marble slab is laid out with squares. The arrangement is such, that each side of a square belongs to two squares and each corner to four squares.

It is a veritable wonder that we can carry out this business without getting into the greatest difficulties. We only need to think of the following. If at any moment three squares meet at a corner, then two sides of the fourth square are already laid, and, as a consequence, the arrangement of the remaining two sides of the square is already completely determined. But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord, then this is an especial favour of the marble slab and of the little rods, about which I can only be thankfully surprised. We must experience many such surprises if the construction is to be successful.

If everything has really gone smoothly, then I say that the points of the marble slab const.i.tute a Euclidean continuum with respect to the little rod, which has been used as a " distance " (line-interval). By choosing one corner of a square as " origin" I can characterise every other corner of a square with reference to this origin by means of two numbers. I only need state how many rods I must pa.s.s over when, starting from the origin, I proceed towards the " right " and then "

upwards," in order to arrive at the corner of the square under consideration. These two numbers are then the " Cartesian co-ordinates " of this corner with reference to the " Cartesian co-ordinate system"

which is determined by the arrangement of little rods.

By making use of the following modification of this abstract experiment, we recognise that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods "

expand " by in amount proportional to the increase of temperature. We heat the central part of the marble slab, but not the periphery, in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not.

With reference to our little rods -- defined as unit lengths -- the marble slab is no longer a Euclidean continuum, and we are also no longer in the position of defining Cartesian co-ordinates directly with their aid, since the above construction can no longer be carried out. But since there are other things which are not influenced in a similar manner to the little rods (or perhaps not at all) by the temperature of the table, it is possible quite naturally to maintain the point of view that the marble slab is a " Euclidean continuum."

This can be done in a satisfactory manner by making a more subtle stipulation about the measurement or the comparison of lengths.

But if rods of every kind (i.e. of every material) were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab, and if we had no other means of detecting the effect of temperature than the geometrical behaviour of our rods in experiments a.n.a.logous to the one described above, then our best plan would be to a.s.sign the distance one to two points on the slab, provided that the ends of one of our rods could be made to coincide with these two points ; for how else should we define the distance without our proceeding being in the highest measure grossly arbitrary ? The method of Cartesian coordinates must then be discarded, and replaced by another which does not a.s.sume the validity of Euclidean geometry for rigid bodies.* The reader will notice that the situation depicted here corresponds to the one brought about by the general post.i.tlate of relativity (Section 23).

Notes

*) Mathematicians have been confronted with our problem in the following form. If we are given a surface (e.g. an ellipsoid) in Euclidean three-dimensional s.p.a.ce, then there exists for this surface a two-dimensional geometry, just as much as for a plane surface. Gauss undertook the task of treating this two-dimensional geometry from first principles, without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. If we imagine constructions to be made with rigid rods in the surface (similar to that above with the marble slab), we should find that different laws hold for these from those resulting on the basis of Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates in the surface.

Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemman of treating multi-dimensional, non-Euclidean continuum. Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.

GAUSSIAN CO-ORDINATES

According to Gauss, this combined a.n.a.lytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig. 4) drawn on the surface of the table. These we designate as u-curves, and we indicate each of them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in the diagram. Between the curves u= 1 and u= 2 we must imagine an infinitely large number to be drawn, all of which correspond to real numbers lying between 1 and 2. fig. 04 We have then a system of u-curves, and this "infinitely dense" system covers the whole surface of the table. These u-curves must not intersect each other, and through each point of the surface one and only one curve must pa.s.s.

Thus a perfectly definite value of u belongs to every point on the surface of the marble slab. In like manner we imagine a system of v-curves drawn on the surface. These satisfy the same conditions as the u-curves, they are provided with numbers in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of u and a value of v belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point P in the diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring points P and P1 on the surface then correspond to the co-ordinates

P: u,v

P1: u + du, v + dv,

where du and dv signify very small numbers. In a similar manner we may indicate the distance (line-interval) between P and P1, as measured with a little rod, by means of the very small number ds. Then according to Gauss we have

ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2

where g[11], g[12], g[22], are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g[11], g[12] and g[22], determine the behaviour of the rods relative to the u-curves and v-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the u-curves and v-curves and to attach numbers to them, in such a manner, that we simply have :

ds2 = du2 + dv2

Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian coordinates are samply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an a.s.sociation of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are a.s.sociated with neighbouring points " in s.p.a.ce."

So far, these considerations hold for a continuum of two dimensions.

But the Gaussian method can be applied also to a continuum of three, four or more dimensions. If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way. With every point of the continuum, we a.s.sociate arbitrarily four numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."

Adjacent points correspond to adjacent values of the coordinates. If a distance ds is a.s.sociated with the adjacent points P and P1, this distance being measurable and well defined from a physical point of view, then the following formula holds:

ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] ... . g[44]dx[4]^2,

where the magnitudes g[11], etc., have values which vary with the position in the continuum. Only when the continuum is a Euclidean one is it possible to a.s.sociate the co-ordinates x[1] . . x[4]. with the points of the continuum so that we have simply

ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

In this case relations hold in the four-dimensional continuum which are a.n.a.logous to those holding in our three-dimensional measurements.

However, the Gauss treatment for ds2 which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble slab of the table and local variation of temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behaviour of the rods is almost as it ought to be according to the rules of Euclidean geometry. Hence the imperfections of the construction of squares in the previous section do not show themselves clearly until this construction is extended over a considerable portion of the surface of the table.

We can sum this up as follows: Gauss invented a method for the mathematical treatment of continua in general, in which "

size-relations " (" distances " between neighbouring points) are defined. To every point of a continuum are a.s.signed as many numbers (Gaussian coordinates) as the continuum has dimensions. This is done in such a way, that only one meaning can be attached to the a.s.signment, and that numbers (Gaussian coordinates) which differ by an indefinitely small amount are a.s.signed to adjacent points. The Gaussian coordinate system is a logical generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua, but only when, with respect to the defined "size" or "distance," small parts of the continuum under consideration behave more nearly like a Euclidean system, the smaller the part of the continuum under our notice.

THE s.p.a.cE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A EUCLIDEAN CONTINUUM

We are now in a position to formulate more exactly the idea of Minkowski, which was only vaguely indicated in Section 17. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, s.p.a.ce-time continuum. We called these " Galileian co-ordinate systems." For these systems, the four co-ordinates x, y, z, t, which determine an event or -- in other words, a point of the four-dimensional continuum -- are defined physically in a simple manner, as set forth in detail in the first part of this book. For the transition from one Galileian system to another, which is moving uniformly with reference to the first, the equations of the Lorentz transformation are valid. These last form the basis for the derivation of deductions from the special theory of relativity, and in themselves they are nothing more than the expression of the universal validity of the law of transmission of light for all Galileian systems of reference.

Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body K by the s.p.a.ce co-ordinate differences dx, dy, dz and the time-difference dt. With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are dx1, dy1, dz1, dt1. Then these magnitudes always fulfil the condition*

dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.

The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude

ds2 = dx2 + dy2 + dz2 - c^2dt2,

which belongs to two adjacent points of the four-dimensional s.p.a.ce-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1], x[2], x[3], x[4], we also obtaill the result that

ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

is independent of the choice of the body of reference. We call the magnitude ds the " distance " apart of the two events or four-dimensional points.

Thus, if we choose as time-variable the imaginary variable sq. rt. -I . ct instead of the real quant.i.ty t, we can regard the s.p.a.ce-time contintium -- accordance with the special theory of relativity -- as a ", Euclidean " four-dimensional continuum, a result which follows from the considerations of the preceding section.

Notes

*) Cf. Appendixes I and 2. The relations which are derived there for the co-ordlnates themselves are valid also for co-ordinate differences, and thus also for co-ordinate differentials (indefinitely small differences).

THE s.p.a.cE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A ECULIDEAN CONTINUUM

In the first part of this book we were able to make use of s.p.a.ce-time co-ordinates which allowed of a simple and direct physical interpretation, and which, according to Section 26, can be regarded as four-dimensional Cartesian co-ordinates. This was possible on the basis of the law of the constancy of the velocity of tight. But according to Section 21 the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the velocity of light must always depend on the co-ordinates when a gravitational field is present. In connection with a specific ill.u.s.tration in Section 23, we found that the presence of a gravitational field invalidates the definition of the coordinates and the ifine, which led us to our objective in the special theory of relativity.

In view of the resuIts of these considerations we are led to the conviction that, according to the general principle of relativity, the s.p.a.ce-time continuum cannot be regarded as a Euclidean one, but that here we have the general case, corresponding to the marble slab with local variations of temperature, and with which we made acquaintance as an example of a two-dimensional continuum. Just as it was there impossible to construct a Cartesian co-ordinate system from equal rods, so here it is impossible to build up a system (reference-body) from rigid bodies and clocks, which shall be of such a nature that measuring-rods and clocks, arranged rigidly with respect to one another, shaIll indicate position and time directly. Such was the essence of the difficulty with which we were confronted in Section 23.

But the considerations of Sections 25 and 26 show us the way to surmount this difficulty. We refer the fourdimensional s.p.a.ce-time continuum in an arbitrary manner to Gauss co-ordinates. We a.s.sign to every point of the continuum (event) four numbers, x[1], x[2], x[3], x[4] (co-ordinates), which have not the least direct physical significance, but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner. This arrangement does not even need to be of such a kind that we must regard x[1], x[2], x[3], as "s.p.a.ce" co-ordinates and x[4], as a " time "

co-ordinate.